On the variation with respect to a parameter of zeros of Bessel and q- Bessel functions
We apply what is essentially the Hellmann-Feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions. These representations prove that the zeros are monotone functions of the parameters involved. This family includes the Bessel functions \(J_{\nu}(x)\) and their basic analog \(J_{\nu}^{(2)}(x;q)\). As a by- product we show that \(j^ 2_{\nu,1}/(\nu +1)\) increases with \(\nu\) when \(\nu\in (-1,\infty)\), \(j_{\nu,1}\) being the smallest positive zero of \(J_{\nu}(x)\). We also use the Lommel polynomials to derive improved lower bounds for the smallest positive zero of \(J_{\nu}(x)\).
scientific article; zbMATH DE number 4061652
scientific article